To determine which set of values could be the side lengths of a [tex]$30-60-90$[/tex] triangle, we need to know the characteristic ratio of the sides in such a triangle.
In a [tex]$30-60-90$[/tex] triangle:
- The side opposite the [tex]$30^\circ$[/tex] angle is the shortest side, denoted as [tex]\(x\)[/tex].
- The side opposite the [tex]$60^\circ$[/tex] angle is [tex]\(\sqrt{3}\)[/tex] times the shortest side, denoted as [tex]\(x\sqrt{3}\)[/tex].
- The hypotenuse (opposite the [tex]$90^\circ$[/tex] angle) is twice the shortest side, denoted as [tex]\(2x\)[/tex].
The ratio of the sides in a [tex]$30-60-90$[/tex] triangle is therefore [tex]\(1 : \sqrt{3} : 2\)[/tex].
Let's examine each option and check if they fit the ratios [tex]\(1 : \sqrt{3} : 2\)[/tex]:
### Option A: [tex]\(\{6, 6\sqrt{2}, 12\}\)[/tex]
1. Shortest side: [tex]\(6\)[/tex]
2. Side opposite [tex]$60^\circ$[/tex]: [tex]\(6 \cdot \sqrt{2}\)[/tex]
3. Hypotenuse: [tex]\(12\)[/tex]
\- Calculating the ratios:
[tex]\[
\frac{6\sqrt{2}}{6} = \sqrt{2}
\][/tex]
[tex]\[
\frac{12}{6} = 2
\][/tex]
The ratios are [tex]\(1 : \sqrt{2} : 2\)[/tex], which does not match [tex]\(1 : \sqrt{3} : 2\)[/tex].
### Option B: [tex]\(\{6, 12, 12\sqrt{2}\}\)[/tex]
1. Shortest side: [tex]\(6\)[/tex]
2. Side opposite [tex]$60^\circ$[/tex]: [tex]\(12\)[/tex]
3. Hypotenuse: [tex]\(12 \cdot \sqrt{2}\)[/tex]
\- Calculating the ratios:
[tex]\[
\frac{12}{6} = 2
\][/tex]
[tex]\[
\frac{12\sqrt{2}}{6} = 2\sqrt{2}
\][/tex]
The ratios are [tex]\(1 : 2 : 2\sqrt{2}\)[/tex], which does not match [tex]\(1 : \sqrt{3} : 2\)[/tex].
### Option C: [tex]\(\{6, 12, 12\sqrt{3}\}\)[/tex]
1. Shortest side: [tex]\(6\)[/tex]
2. Side opposite [tex]$60^\circ$[/tex]: [tex]\(12\)[/tex]
3. Hypotenuse: [tex]\(12 \cdot \sqrt{3}\)[/tex]
\- Calculating the ratios:
[tex]\[
\frac{12}{6} = 2
\][/tex]
[tex]\[
\frac{12\sqrt{3}}{6} = 2\sqrt{3}
\][/tex]
The ratios are [tex]\(1 : 2 : 2\sqrt{3}\)[/tex], which does not match [tex]\(1 : \sqrt{3} : 2\)[/tex].
### Option D: [tex]\(\{6, 6\sqrt{3}, 12\}\)[/tex]
1. Shortest side: [tex]\(6\)[/tex]
2. Side opposite [tex]$60^\circ$[/tex]: [tex]\(6\sqrt{3}\)[/tex]
3. Hypotenuse: [tex]\(12\)[/tex]
\- Calculating the ratios:
[tex]\[
\frac{6\sqrt{3}}{6} = \sqrt{3}
\][/tex]
[tex]\[
\frac{12}{6} = 2
\][/tex]
The ratios are [tex]\(1 : \sqrt{3} : 2\)[/tex], which matches [tex]\(1 : \sqrt{3} : 2\)[/tex].
Hence, the correct set of values that could be the side lengths of a [tex]$30-60-90$[/tex] triangle is:
[tex]\[ \boxed{D \{6, 6\sqrt{3}, 12\}} \][/tex]
